As @filoxo said, you can use document.querySelectorAll.
If you know that there is only one element with the class you are looking for, or you are interested only in the first one, you can use:
document.querySelector('.class1.class2');
BTW, while .class1.class2 indicates an element with both classes, .class1 .class2 (notice the whitespace) indicates an hierarchy - and element with class class2 which is inside en element with class class1:
<div class='class1'>
<div>
<div class='class2'>
:
:
And if you want to force retrieving a direct child, use > sign (.class1 > .class2):
<div class='class1'>
<div class='class2'>
:
:
For entire information about selectors:
https://www.w3schools.com/jquery/jquery_ref_selectors.asp